# What is a subset?

Technical Answer: Set A is a of a Set B if and only if elements of A are also elements of B
In Other Words ...
There are two ways to be a subset
1) All the elements in Set A in Set B
OR
2) Set A has the elements as B

## Subsets in Pictures

 All elements of Set A are contained in Set B : Set A has the exact same elements as Set B

## Subset Notation

$\subseteq$ is the symbol for subset
 Notation Meaning A $\subseteq$ B "A is the subset of B" B $\subseteq$ A "B is the subset of A"

## Examples

 1) Is Set A = { 2,4,6} a subset of B = {1, 2 , 3, 4, 5, 6 } Yes no
 2) Is Set A = {10, 15,20} a subset of B = {10,20,30,40 } Yes no
 3) Is $A=\{x: x \in \mathbb{Q}\}$ a subset of $B=\{x=\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots\}$? Yes no
 4) Is $A=\{x: x \in \mathbb{R}\}$ a subset of $B=\{x: x \in \mathbb{Q}\}$? Yes no
 5) Is set P = {} a subset of set M ={0,1,4,9,25,36}? Yes no
 6) Is $A=\{x: x \in \mathbb{Z^+}\}$ a subset of $B=\{x: x \in \mathbb{N}\}$? Yes no
 7) Is $A=\{x: x \in \mathbb{N}\}$ a subset of $B=\{x: x \in \mathbb{Z^+}\}$? Yes no

## Key Ideas

I. a set is a subset of itself
II. $\varnothing$ is a of every set
III. If A $\subseteq$ B and B $\subseteq$ A then
 Based on this last definition are A = {1,2,3,4,5,6,7 } and B = {x|x< 7 and $x \in \mathbb{N} \$} equal? Yes no

## Practice Problems

 1) Is {y } $\subseteq$ {y}? Yes no
 2) Is Set A = { y } $\in Set B=$ {y {y} }? Yes no
 3) Is Set A = {y} $\subseteq Set B=$ { y , {y} }? Yes no